The Determination of Rent

A Discussion of the Mathematics
and Theoretical Assertions Involved

Fred Foldvary, David Hillary and Dan Sullivan

[Reprinted from a Land-Theory online discussion, October 2000]

DAN SULLIVAN

Fred makes a shift in the meaning of growth rate.

David Hillary responds: Please note that I have always used g to be the rate of growth of (gross) rent, as defined in my proof. Fred appears to have changed his tune, I have not.

DAN SULLIVAN

The obvious solution to making the formula work properly is to define g as a growth in *net* rent, for it is net rent, not gross rent, that interests the investor. The price will, in turn, be projected from the expected gain in net rent. Of course, this is what David called "bullshit," but the numbers work.

David Hillary responds: Earlier in this post (whoops deleted it) you stated that it was wrong to employ in a function an argument which depended on the function, as a solution for stating the function's value. There is a grain of truth in this assertion. However in the above part of your post you state that the rate of growth of the net rent should be used in the value function. However with value taxation, the net rent depends on the value through time. If the price-rent ratio is constant and the rent growth rate is constant this may be admissable. If you want to prove a land value function you will need to define value in an acceptable way and move from this definition of value to your final formula using valid steps. I define value of an asset as being equal to the sum of the discounted expected net income. Then I express the income in terms of gross rent less tax in each period. Then I solve for the value. i.e.: Pn=f(Ii)=sum{i=n to inf}((Ii)*(1+d)^(n-i-1)) Ii=Ri-Ti Ri=R0*(1+g)^i Ti=t*Pi and so on to get d=y-t+g at the end.

DAVID HILLARY

Dan, it's time you conformed to some mathematical and logical conventions such as stating your premesis and using a valid argument form. Until then its hard to refrain from describing your claims as "complete bullshit."

DAN SULLIVAN

Actually, Fred is right that Solow's formula works if we redefine "g", but we must redefine it not from growth in rent to growth in price, but from growth in gross rent to growth in net rent. After all, just as the current investor is interested in net rent, not gross rent, so is the price dependent on what future investors will be interested in, and they, too, will be interested in net rent, not gross rent. This is where David went astray initially. He projected a perpetual growth in price based on a perpetual growth in gross rent, even though he figured the tax collectors to be collecting the entire rent. Hence, the net rent, being zero, meant that the 5% growth in net rent is impossible, for growth is the new rent divided by the old rent, and anything divided by zero (other than zero itself) is infinity.

DAN SULLIVAN

This does make the numbers come out, as we shall see in Fred's new, improved formulation.

Fred Foldvary responds: For example, at time T=1, r=100, t=.2, g=.05, i=.05 p=100/(.05+.2-.05)=100/.2=500

DAN SULLIVAN

Earlier, [Fred Foldvary] said, "g is not a function of p. g is an independent variable empirically observed, as is interest rate i." That is false. Both g and i are predictions to infinity as to what both rent growth and interest rates will be, and therefore absurd on its face, for nobody can state with confidence what either g or i will be even in 10 years. But if we are supposing, rather than predicting, we really should be supposing based on something, and the growth in price is predicated on an expected growth in net rents.

FRED FOLDVARY

In my example of g=.05, p is empirically observed to be growing at 5%.

Dan Sullivan responds: No, you are predicting a 5% growth for the future, not observing one from the past. But, we will pretend you can do that, so long as your results are consistent.

FRED FOLVARY

One year later, T=2, r=105, and p=105/.2=525 as both rent and price have risen by 5%.

Dan Sullivan responds: But net rent is zero, and so is the growth in net rent zero. Therefore, you are heading for error, as we shall see.

FRED FOLDVARY

p and r will keep growing at 5% until something changes.

Dan Sullivan responds: But net rent will be zero until something changes, according to your equation. This will prove important, as there can be no percentage growth in zero.

FRED FOLDVARY

Now let's move to 20 years later. At T=20 r=271; p=271/.2=1359 But suddenly and unexpectedly the rent stops growing. So far the title holder has not actually gotten any rent payment, because all the current rent was taxed. But the expectation was that someday, if the rent stopped growing, then the title holder would get 20% of the rent, and that would be a larger rent than the present-day rent.

Dan Sullivan responds: You have contradicted yourself, in saying that the rent stopped unexpectedly, but that the expectation was that someday, if the rent stopped growing, then the title holder would get 20% of the rent. Thus, the only way the land is worth *anything* is if the rent stops growing. (If it slows growing, he will get perhaps 10% of the rent, but that just muddies the distinctions.)

FRED FOLDVARY

So if g falls to zero, then at T=20 p=271/(.05+.2)=271/.25= 1084. The price falls from 1359 to 1084 because the growth of rent and price are now zero.

Dan Sullivan responds: But the person who bought the land originally (or the one who paid 1359) still made a sucker bet, because either the land would never yeild rent or it would stop growing and collapse the price. There is no other way.

FRED FOLDVARY

The percentage gain, after g falls to zero, is now log(1084/500)/20=.039 or 3.9%.

Dan Sullivan responds: In other words, the person expecting a 5% return paid about 20% too much, and/or was assessed about 20% too high. Not coincidentally, Victor's assertion was that about 20% of the rent would be left with the landholder, and a 20% assessment reduction would accomplish that.

FRED FOLDVARY

The problem is, nobody knew when the rent and price would stop growing.

Dan Sullivan responds: It doesn't matter, because there is no way the return can come out right regardless of what year, as we shall see from your 100 years example.

FRED FOLDVARY

If they knew it would stop growing in 20 years, the formula would not be p=r/(i+t-g) but a present value only for the rents to be received starting in 20 years.

Dan Sullivan responds: A better formulation (assuming your price assessment, which will prove to have been too high) is to say that the net rents are zero for 20 years and become $271 on the 21st year and thereafter. Thus, the total return after 20 years is $1084, minus the $500 originally invested, for a yeild of $584 profit. Had the person paid $408, $1084 would have been a marginal return. Once again, you are off by just under 20%. (This pattern should serve as a hint.)

FRED FOLDVARY

So why would someone pay for land at r/(i+t-g) if he is collecting no rent and if when rent stops growing the percentage gain will be less? Because nobody knows when the rent will stop growing, and they expect the rent to keep growing indefinitely. Suppose it grows for 100 years. Then r=14,841 p=74,207 when g becomes 0, then p=59,365, with a growth rate of 4.77%

Dan Sullivan responds: I don't know where you got those numbers. The price, after 100 years of growth at 5%, should be $500 * 1.05^100, which is 65,750, and the net rent to sustain such a price would have to be 5% of that, or 3,287 per year. This means that, for the capitalized value to have actually risen by 5%, the rent would have to rise by 5.235% If the rents rose 5%, as you are predicating, *gross* rent becomes $13,150 per year, which is $100 * 1.05^100. The net rent becomes $2630. The price, which is the rent divided by the interest rate, becomes $52,600, or 80% of the return he could have gotten by investing at the interest margin.

DAN SULLIVAN

Suppose it grows for 100 years. Then r=14,841 p=74,207 when g becomes 0, then p=59,365, with a growth rate of 4.77%

Fred Foldvary responds: I don't know where you got those numbers. I'm using continuous compounding: F=P*e^(it) it=5 since i=.05 and t=100 so F=500*exp(1)^5

DAN SULLIVAN

The price, after 100 years of growth at 5%, should be $500 * 1.05^100, which is 65,750

Fred Foldvary responds: You are using annual compounding, where the rent stays the same for a year and then suddenly jumps up 5%. I am using continuous compounding where the rent grows continuously. For one particular property, the landlord may raise the rent once a year, but for a whole economy, the more likely case is that rents are being increased at different times, and as whole they rise continuously.

DAN SULLIVAN

In that case, the comparison is valid if you use the same methodology to calculate the alternative return of 5% interest, and you will still get the same errors. I just posted a formula that has no such errors. I compounded annually for the rent growth as well as for the comparative marginal investment, but I am sure the end result would be the same if the compounding were continuous in both cases.

FRED FOLDVARY

The price still falls by the same percentage, but because of the long time period, the compounding growth can be closer to 5%. Far enough into the future, a small change in percentage growth becomes a large change in the final amount.

Dan Sullivan responds: No, you did the math wrong somewhere. The return gets closer and closer to 80%. By 100 years, it is so close that a little rounding makes it 80% exactly.

FRED FOLDVARY

So the further into the future the growth continues, the closer to 5% will be the return when the growth stops.

Dan Sullivan responds: Wrong. 4% This seems to be what is happening with internet stocks. They make no current profit, but are expected to keep growing rapidly. But if growth stops in the near future, the return will not be so high. So some of these stocks were priced unrealistically high, as though growth would > continue forever. But it could not be sustained, and that's why the stock values fell so much. What is this, the greater-fool formula?

DAN SULLIVAN

If there is no growth, the formula is easy. Price = net yield / interest rate. or, Price = Rent - Tax / interest rate

David Hillary responds: Once again it becomes necessary for me to demolish a vain attempt to overturn the land value equation d=y-t+g. ...David Hillary's comments are continued below.

DAN SULLIVAN

Here is a formula that works, and is consistent at 1 year, at 20 years, at 100 years, and at any year. But, by all means, let us think about what each part of the formula is actually saying. ... Let us use the numbers Fred used to get zero net rent. That is, the interest rate is .05 the capitilzation rate, k, is therefore 20

Victor Levis: Only when the income stream is steady. Otherwise, use of this term will confuse, and not illuminate.

VICTOR LEVIS

Only when the income stream is steady. Otherwise, use of this term will confuse, and not illuminate.

Dan Sullivan responds: The original formula also depended on the income being steady. I have been the most forthright of anyone in saying that no formula can predict real life. However, the other formula was not working even when the income stream *was* presumed to be steady.

DAN SULLIVAN

The original formula also depended on the income being steady. I have been the most forthright of anyone in saying that no formula can predict real life. However, the other formula was not working even when the income stream *was* presumed to be steady.

Victor Levis responds: What I mean by 'steady', is that there is no growth at all. This capitalization rate, or k, is not necessary in these formulae. It only holds true in a special case.

VICTOR LEVIS

What I mean by 'steady', is that there is no growth at all. This capitalization rate, or k, is not necessary in these formulae. It only holds true in a special case.

Dan Sullivan responds: The capitalization rate is nothing more than one over the interest rate. It holds true in any hypothesis where the interest rate holds true.

VICTOR LEVIS

If by capitalization rate you mean the ratio of capitalized value to the income stream, it is ONLY 'one over the interest rate' when the income stream is steady. In the case of 4% growth in income stream, and 5% discount rate, you agree below that the capitalization rate is 100 and not 20.

DAN SULLIVAN

The k constant in my formula was simply the inverse of the discount rate or prevailing interest rate. It was applied, in my formula, to the entire return, which means growth value plus rent. Thus, it would still be 20, but 20 * the entire yield, which is growth value plus rent, no just rent.

Victor Levis responds: Then we have just found your error. There is no capitalization factor of 20. You need to do a Net Present Value on each and every year's worth of net rent. The k=1/i only works if the income stream never changes, and it is a convenient shorthand-trick, not something to rely on in more complex cases. I remember you saying something like "the entire yield is $19.23 plus 5%" but that cannot be right, as that is only the second year yield, and does not fully take into account the permanent growth engendered by the 5% increase. Only doing a separate NPV for each year takes everything into account.

DAN SULLIVAN

The growth rate, g, is also .05, which means the growth factor is 1.05

Victor Levis responds: Each and every year.

VICTOR LEVIS

Each and every year.

Dan Sullivan responds: According to the flawed formula, yes. I am only offering a formula that is *mathematically* correct.

DAN SULLIVAN

According to the flawed formula, yes. I am only offering a formula that is *mathematically* correct.

Victor Levis responds: Accepted. Unfortunately, while I sympathize with what you are trying to do, and especially in how you are trying to get David to stop shouting at us and to finally recognize that no one will pay $25,000 for a piece of land that carries no net rent, your formula also goes wrong, but in a different place.

DAVID HILLARY

Once again it becomes necessary for me to demolish a vain attempt to overturn the land value equation d=y-t+g.

Dan Sullivan responds: Translation: I AM THE GREAT AND POWERFUL WHIZZER OF ODDS. PAY NO ATTENTION TO THE MAN BEHIND THE UNCERTAINTIES.

DAVID HILLARY

THIS EQUATION IS CORRECT AND IT WORKS. OTHER PROPOSED EQUATIONS DO NOT WORK UNLESS THEY ARE EQUIVILENT TO IT. THIS CAN BE DEMONSTRATED WITH NUMERIC EXAMPLES. GIVE UP ON OVERTURNING IT.

DAVID HILLARY

R=$100 d=0.05 t=0.2 g=0.05 claimed P $403.85 Actual P $500

Dan Sullivan responds: "Actual P" is merely David's claimed P <

DAVID HILLARY

claimed P for period 21 $1 071.52 Claimed rate of P growth is 0.05 per period. i.e. price growth equals rent growth.

Dan Sullivan responds: Is proportional, not equals, and it is actually proportional to net rent growth. If net rent growth is zero, because net rent is zero, then it doesn't matter what gross rent growth is.

DAVID HILLARY

so P in year 2 is going to be $424.04, a gain of $20.19. Net rent is $100 less 0.2*$403.85 which is $19.23. Total return to the investor is therefore $39.42. This 9.76% of $403.85. The rate of return should be d, which is 5%.

DAVID HILLARY

Dan, your formula is defective, untrue and more complex that it needs to be. d=y-t+g. This is the truth. I will show you. R=$100 t=0.2 d=0.05 g=0.05 so we have P=$100/(0.05+0.02-0.05)=$500 Net rent is $100-$500*0.2=0, capital gain is 0.05*$500, which gives a total return of 0.05*$500,which equates to a 5% rate of return, like it should. There is only one Price-Rent ratio that give a d rate of return in the form of the sum of the capital gain and the net rent, and that is d+t-g.

Dan Sullivan: David, I will go over my formula again, but your formula is clearly wrong for net rent = 0 because there is no actual return, now or ever, to the investor, so long as the parameters of your formula are true. That is, you presume a growth in price, when price is derivative of a growth in net rents, and your scenario posits no growh in net rents. When they are no longer true, everything must be readjusted, as Fred has shown.

DAN SULLIVAN

The question becomes how to define and calculate growth. Clearly, the growth that the investor is looking for is a growth in net returns, not in price per se, and certainly not in gross rent. After all, price is derived from a projected growth in net returns, and not from anything else. After all, we are trying to calculate actual value, not the sucker-price.

DAN SULLIVAN

Let us presume that the grand poobah of prediction has everyone rightly convinced that rents will rise by 5% indefinitely, which just happens to be the interest rate. Now let us logically construct a formula.

DAN SULLIVAN

Price is the capitalization of expected annual yield plus the capitalization of expected growth in that yield.

DAN SULLIVAN

Let us call k the capitalization rate, which is the inverse of the interest rate. Thus, if the interest rate is .05, k is 20. Not a big deal; it just saves repetition of a step. Thus, P = k*(r-t+g) Except that, first of all, the tax is a portion of the price, and not just a number. Hence it is cleaner to write, P = k*(r-tP+g)

DAN SULLIVAN

But what is g? If it is not a growth in the net yield, then it is irrelevant to the investor, or to the future investors on whom he will attempt to unload the property. The better expression to capture g, then, is to say it is a factor of r-tP. Thus we have, P = k *[(r-tP)+ g(r-tP)] That is, Price = K * [(rent-tax on price)+growth(rent-tax on price)] Factoring out the r-tP, we get, P = K [(1+g)(r-tP))

DAN SULLIVAN

Now, the nature of growth, or compounded interest, is that you don't deal with the interest alone, but you multiply the principle by itself with interest each year. That is, if something has a 5% growth rate, what actually compounds is the value itself, which becomes 1.05 times the prior value. So, above, "1+g" is actually a more cogent number than g itself.

DAN SULLIVAN

To elaborate, rather than saying something gains 5% interest each year, it is mathematically cleaner, and less prone to error, to say that it multiplies by 1.05. Indeed, the erroneous formula was adding growth, with no sense of growth to *what*. Growth is a multiplying effect, and so it is much better to have a new number G, which is (1+g). This also helps with negative interest, as G is still positive. Just keeping things simple so the "tyros" won't get confused. :^) P = k*G(r-tP)

DAN SULLIVAN

I see the tax is a function of the price, which means we have price on both sides of the equation. Not good, but that is one reason assessing the tax against the price is not such a good idea. Anyhow, it is easy enough to fix after numbers are plugged in. ...

DAN SULLIVAN

Let us use the numbers Fred used to get zero net rent. That is, the interest rate is .05 the capitalization rate, k, is therefore 20 The growth rate, g, is also .05, which means the growth factor is 1.05

Victor Levis responds: Only when the income stream is steady. Otherwise, use of this term will confuse, and not illuminate.

DAN SULLIVAN

The initial rent is $100.

Victor Levis responds: OK

DAN SULLIVAN

... the tax rate is 20% of the price, or .2; the tax itself is .2P

Victor Levis responds: OK. Or more precisely .2 of assessed value.

VICTOR LEVIS

OK. Or more precisely, .2 of assessed value.

Dan Sullivan responds: No, it's .2 of market value. It could be .4 of assessed value, if the assessment is half the market value.

DAN SULLIVAN

No, it's .2 of market value. It could be .4 of assessed value, if the assessment is half the market value.

Victor Levis responds: Whatever. The fact is, it's based on an assessment, and all of you are only ASSUMING that the assessment has a firm connection to the price. The investor does not really care about anything except the expected tax value in dollars for each year.

VICTOR LEVIS

Whatever. The fact is, it's based on an assessment, and all of you are only ASSUMING that the assessment has a firm connection to the price. The investor does not really care about anything except the expected tax value in dollars for each year.

Dan Sullivan responds: These formulas are all about assuming what cannot be assumed. However, the effect of tax collection on value derives from the tax as a percentage of the *actual* value. That is, if they officially assess at 100% but in fact assess this property at 80%, then a 25% tax on paper is a 20% tax in fact. In any case, the formula depends on the tax in fact, and assumes that ratio of the assessment to market value will, like everything else in the formula, remain constant.

DAN SULLIVAN

These formulas are all about assuming what cannot be assumed. However, the effect of tax collection on value derives from the tax as a percentage of the *actual* value. That is, if they officially assess at 100% but in fact assess this property at 80%, then a 25% tax on paper is a 20% tax in fact. In any case, the formula depends on the tax in fact, and assumes that ratio of the assessment to market value will, like everything else in the formula, remain constant.

Victor Levis responds: Let me repeat MY calculation for growth of 4% annually with a discount rate of 5%.

VICTOR LEVIS

Let me repeat MY calculation for growth of 4% annually with a discount rate of 5%.

Dan Sullivan responds: Sure. Now that we are not dividing by zero, it is a mathematically realistic scenario. (However, it still yields an unrealistically low ratio of rent to price, and so I think 4% growth in rents is well beyond reality.)

VICTOR LEVIS

Let me repeat MY calculation for growth of 4% annually with a discount rate of 5%. Show me ANY flaw with its determination of the land's market value.

Dan Sullivan responds: For the value to be a normal investment value, it has to promise a normal return, which is the combination rent and growth.

DAN SULLIVAN

For the value to be a normal investment value, it has to promise a normal return, which is the combination rent and growth.

Victor Levis responds: Agreed. Unfortunately, you didn't calculate the growth correctly.

VICTOR LEVIS

Agreed. Unfortunately, you didn't calculate the growth correctly.

Dan Sullivan responds: Probably not. Did you try it with numbers from a valid scenario? I was just wondering if my formula was shown to be invalid for the same reason that David's was.

DAN SULLIVAN

To be more precise, the combined return left to the investor from net rent plus growth of net rent will be exactly 80% of the price, and the bulk of that will be rent itself.

DAN SULLIVAN

Now to solve. P = k * G * (r-tP) = 20 * 1.05 * ($100-.2P) = 21 * ($100 - .2P) P = $2100 - 4.2P 5.2P = $2100 P = 403.85

Victor Levis responds: I'll check your result and see if that makes sense.

DAN SULLIAN

Now that we have solved for P, let us see how the numbers play out in year one, year 20, and year 100. Year one. Rent = 100 Price = $403. Tax = .2P = $80.77 Net rent = $19.23 Growthfactor = 1.05* net rent = $20.19

Victor Levis responds: What is this 'growth' you are talking about?

DAN SULLIVAN

That is, rent plus growth = $20.19 That just happens to be exactly 5% of the price.

Victor Levis responds: It also just happens to be a meaningless figure. Well, not completely meaningless. It is the net rent of year 2. But just year 2. In year 3 it goes higher still. And so forth.

VICTOR LEVIS

The way I understand it, it is this: In year one, the net rent is $19.23, because the assessor has valued the property at $403.85 and the council has levied a tax of 20%, thus hitting the landowner with $80.77 of tax against $100 of rent.

VICTOR LEVIS

In year 2, the net rent will go up by 5%, because the gross rent will go up by 5%, and so will the assessment. Therefore, in year 2, the net rent will be $20.19. In year 3, the net rent will be $21.20.

VICTOR LEVIS

What is the net present value of the whole stream? It is the sum of the net present values of each individual year. The year one NPV, assuming the tax is paid at the end of the year, the same time as the rent is received, is $19.23 divided by 1.05 equaling $18.31. The year two NPV is $20.19 divided by 1.05 divided by 1.05 again, which is $18.31 again.

VICTOR LEVIS

If you do the math, you will find that the NPV is $18.31 each and every year, and so the sum of the NPV's is.........infinite. That's right. Infinite. The assessors have under-assessed.

David Hillary responds: Remember that an increase in valuation will increase the tax and therefore reduce the net rent, which means an increase in valuation (in terms of Price Rent ratio) will ultimately eliminate the infinite value.

DAVID HILLARY

Remember that an increase in valuation will increase the tax and therefore reduce the net rent, which means an increase in valuation (in terms of Price Rent ratio) will ultimately eliminate the infinite value.

Victor Levis: Yes, I did assume that Dan was doing the Assessing, and increasing the assessment by 5% each year. :-)

VICTOR LEVIS

Which is why the discount rate for land cannot be permanently equal to or less than the expected growth rate in rent.

David Hillary responds: You have proved that the assessment is incorrect. Dan's like that, he just can't figure out how to do valuations of income streams properly. You are correct to say that at $403 or whatever Dan give is under-assessed. The correct value is $500. You are incorrect to say that the rent growth rate cannot equal the discount rate perpetually. In Singapore the real economic growth rate averages around 6-7% and the real interest rate is between 1 and 2%.

DAVID HILLARY

Current estimated sustainable economic growth rates in rich capitalist countries are about equal to or in excess of the world real interest rate and their own domestic interest rate. If taxation and public pensions were reduced or eliminated growth would be higher and savings would be higher and the interest rate might be even lower. Rent seems to be directly connected to overall economic activity and rent growth and economic growth will be very similar to each other.

DAVID HILLARY

The solow growth model indicates that the economic growth will always equal the interest rate if the consumption maximising savings rate occurs. There is no reason to think that d cannot equal g on an ongoing basis.

VICTOR LEVIS

Which is why the discount rate for land cannot be permanently equal to or less than the expected growth rate in rent. You have proved that the assessment is incorrect. Dan's like that, he just can't figure out how to do valuations of income streams properly. You are correct to say that at $403 or whatever Dan give is under-assessed. The correct value is $500. If the tax takes all the rent, and is expected to do so forever, then Fred and I are right, the private value is 0. The public value, the value to the community as a whole, since it owns all the rent, will be finite only if the expected growth in rent is less than the discount rate.

David Hillary responds: From what I could gather Fred thinks the private value is $500, as I did.

DAVID HILLARY

From what I could gather Fred thinks the private value is $500, as I did.

Victor Levis responds: But he specifically stated that this was because it was not expected for rent to rise indefinitely. This is the statement we are waiting for you to make, as well.

VICTOR LEVIS

You are incorrect to say that the rent growth rate cannot equal the discount rate perpetually. In Singapore the real economic growth rate averages around 6-7% and the real interest rate is between 1 and 2%. So, do people pay INFINITE amounts to buy the tiniest piece of land? Or do they pay finite amounts? If they pay finite amounts, then Fred is probably right, and land investors do NOT believe that the their land rent will grow 6-7% every year, indefinitely, no matter what YOU believe or say is 'sustainable'. In fact, I don't understand why you haven't put every single dollar you own into buying Singapore real estate. Let me give you a clue into what may be astray in your analysis. You started the above paragraph saying that I am wrong about the rent growth rate, but ended it with an empirical observation of economic growth, not rent growth. As I said the other day, I retract by statement that economic growth cannot exceed the discount rate. It can, so long as this growth is not capital-dependent. But if and when such a phenomenon ever exists, despite the assumptions of the Solow model, rent is not in fact expected to increase by as much as the overall economy.

David Hillary responds: ok we have four variables here: i the interest rate in the econony (net marginal product of capital in SGM) d the discount rate on land, equal to i plus p, the risk premium on land, which depends on how risky the land asset is. (I have modelled the risk premium in my macro-model as being proportional to the sensitivity of the land value to the change in d-g) g the rent growth rate the other variable is the growth of output. I have used assumed that rent is a fixed fraction of output, based on the use of a form of Cobb-Douglas production function (Y=th*L^a*K^b*N^(1-a-b) where th is technology, L is the labour force employed, K is the capital stock and N is the stock of land (fixed). a and b are constants. Factor incomes are their marginal products giving labour a*Y, capital b*Y and land the remainder. Given the difficulty in measuring factor shares because of indirect taxes and the business cycle and so on, it seems that the wage share of output does go up and down over time. However the Cobb-Douglas production function was created on the basis that in the long term the factor shares seemed to be unchanged over long periods of time. I think that for the type of modelling we are doing this approach is the best available and that treating rent as a fixed share of output is sensible and believable, despite being a simplification of reality. More advanced modelling of land endowment and use could yield different results, but i suspect it would raise very substantial difficulties in using for the aggregate valuation of land and the effect of land value tax on land value.